Polynomial Growth Harmonic Functions on Finitely Generated Abelian Groups

TL;DR

This paper develops geometric analytic methods to study polynomial growth harmonic functions on Cayley graphs of finitely generated abelian groups, providing new proofs and precise dimension calculations that depend solely on the group.

Contribution

It offers a geometric analytic proof of classical theorems and computes the exact dimension of polynomial growth harmonic functions for finitely generated abelian groups.

Findings

Provided a new geometric proof of Heilbronn and Nayar theorems.

Calculated the exact dimension of polynomial growth harmonic functions.

Showed the dimension depends only on the group, not the generating set.

Abstract

In the present paper, we develop geometric analytic techniques on Cayley graphs of finitely generated abelian groups to study the polynomial growth harmonic functions. We develop a geometric analytic proof of the classical Heilbronn theorem and the recent Nayar theorem on polynomial growth harmonic functions on lattices $\mathds{Z}^n$ that does not use a representation formula for harmonic functions. We also calculate the precise dimension of the space of polynomial growth harmonic functions on finitely generated abelian groups. While the Cayley graph not only depends on the abelian group, but also on the choice of a generating set, we find that this dimension depends only on the group itself.